What is the Blender equivalent of snapping a vacuum cleaner's attachment onto the main hose?
You can do this with my addon, Mesh Align Plus. GIF below.


Select 3 verts on your fixed destination (the vacuum cleaner), hit 'Grab Destination'
Select the corresponding verts on your hose (the thing you want to move) and manually grab (uncheck auto-grab and hit 'Grab Source')
Select the verts you want to move, and hit apply to: "Mesh Piece" (make sure 'Enable Experimental' is checked, this warns you about non-uniform scaling...apply it if you have non-uniform scaling on your object).




The selection order is very important (it is used to orient the mesh during the transformation), so make sure the corresponding verts are selected in the same order. Your targets don't need to both be within the same mesh, also (object->object, mesh->object etc. are also possible, but slightly different)

It would probably be best to describe this as a 'mate' operation, or plane/face alignment...